Unit 9: Vectors, Matrices and

TransformationsBy: Sushy Balraj and Sonal Verma

★ Any quantity that contains both length and direction.

★ It is named from the initial to the terminal point.

Vectors

2 right

3 up

Name:

➔ Example 1: ⇀

★ Component form shows the horizontal and vertical change of the vector.

❖ Naming vectors are just like naming rays (Unit 1). We name both from the endpoint (initial) to the arrow (terminal).

Component Form: <2,3>

OA

➢ Common Mistake: People forget to put the brackets around the component form of the vector.

Vectors

★ Magnitude of a vector relates to the length of it. To find magnitude use the distance formula:

★ Amplitude is the direction in which the arrow points. To find the amplitude use: SOHCAHTOA.❖ We learned how to find unknown side lengths and angle measurements of triangles using sine, cosine, and tangent (Unit 7 Part 2).

Real Life Example of Vectors

What is the component form for vector AB and vector BC?

BC: <16-12, 2-4> = <4, -2>AB: <12-0, 4-0> = <12,4>

★ Rectangular array of elements.★ Arranged in rows and columns.

Matrices

(3x3), (3x3)

★ In order to add or subtract matrices, the dimensions need to be the same.

★ Example 2:

★ Multiplying matrices…

Matrices

➢ Common Mistake: People often forget the prerequisites to add, subtract and multiply the matrices.

Real Life Example of Matrices

STEP 1 STEP 2

Examples of Matrices

A+B=

(2x4) (4x3)

★ Sliding a figure from one position to another.★ The shape and size stays congruent to the

original figure.

Translations

★ Use motion rules, component forms, or vectors to indicate translation.

(x, y) (x-5, y-2)

★ Units move up (+), down (-), left (-), right (+).

Common Mistake: People forget to put the negative behind the coordinate may lead to an incorrect translation.

Reflections★ Mirror images★ flipped over “line of reflection”

★ Every point of reflection is the same distance from the line of reflection as the corresponding point on the original figure If reflecting over...x-axis: (x,y) (x, -y)

y-axis: (x,y) (-x, y)y=x: (x,y) (y, x)y=-x: (x,y) (-y, -x)

★ Turning an object ★ Direction can be counterclockwise (CCW) or

clockwise (CW), if not specified- then always CCW★ Coordinate Rules-90 CCW: (x,y) (-y, x) [0 -1] [1 0]180 CCW: (x,y) (-x, -y) [-1 0] [0 -1]270 CCW: (x,y) (y, -x) [0 1] [-1 0]360 CCW: (x,y) (x,y) [1 0] (Parent matrix) [0 1]

Rotations

➢ Use matrix multiplication to figure out the points for rotation by multiplying the matrix by the coordinates.

Rotation

➢ Struggle: Memorizing the coordinate rules for rotations.

➢ Common Mistake: 90° CW is actually 270° CCW!

★ stretch or shrink ★ k= scale factor★ enlargement= k>1★ reduction= 0<k<1

Dilations

➔ Example 3: Use the given point and k=2 to dilate the figure. Remember to use a ruler to get precise 📏

measurements!

★ combining transformations★ translation + reflection = glide reflection

Composition of Transformations

➔ Answer: Dilation- multiply everything by 2; A(-4,-4), B(-4, -8), C(-8, -8)Reflection over y-axis: (-x,y); A(4, -4), B(4, -8), C(8, -8) A(4,-4) B(4,-8) C(8, -8)

➔ Example 4: The vertices of triangle ABC are A(-2, -2), B(-2, -4), C(-4, -4). List the coordinates after a composition of transformation.Dilation: centered at the origin with a scale factor of 2Reflection: across the y-axis

★ Different ways to approach transformations

Buried Treasures and Clock Problems

➔ Example 6:Start: (6,-1)Translate: (x+2, y-1)Reflect: x-axisRotate: 180°Component: <-2, 1>Where is the treasure?

➔ Example 5:

Start: 10Rotate: 180°Reflect: x-axis Rotate: 150°CCWWhat time is it?

Now there’s no possible way you

could fail this unit :)